Regularization of central forces with damping in two and three-dimensions

TL;DR

This paper investigates the regularization of damped central force motions in two and three dimensions, deriving mappings to undamped systems and revealing nonlinearities introduced by damping effects.

Contribution

It extends regularization techniques to damped systems with various potentials, including Kepler and harmonic, using Levi-Civita, Kustaanheimo-Stiefel, and Bohlin-Sudman transformations.

Findings

Regularized equations are nonlinear due to damping.

Damped Kepler motion maps to an oscillator with inverted sextic potential.

Harmonic oscillator with damping maps to an undamped system with shifted frequency.

Abstract

Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation.